A two-cube cubulation of the AIM free-by-cyclic example

Abstract

We construct a finite locally CAT(0) unit cube complex with one vertex, seven edges, eight squares, and two three-dimensional cubes, whose fundamental group is the mapping torus of a→ a, b→ ab, c→ bcb. Its universal cover gives a proper cocompact cubical action of the exact group in AIM Question 5.2. All attaching maps, signed vertex links, and reversible presentation changes are given explicitly. We also explain a qualitative existence deduction from earlier WPD, Morse, and cubical completion theorems. The contribution presented here is the explicit small finite model; no historical firstness or minimality claim is made.

The idea in plain English

Some groups can be understood through the geometry of spaces assembled from squares and cubes. A useful arrangement has nonpositive curvature and repeats so that a finite quotient describes the entire infinite space.

This paper gives an explicit construction for one group in the AIM problem list. Its finite model has one vertex, seven edges, eight squares and two three-dimensional cubes. The face identifications and the resulting fundamental group are specified. The universal cover is the infinite CAT(0) cube complex on which the group acts freely, properly and cocompactly.

The compact description is the interesting feature. “Two cubes” counts the three-dimensional cells in a quotient that allows boundary identifications. It does not mean that every point on the boundaries of two ordinary solid cubes remains distinct.

The precise result

Let F be the free group on a, b and c, with automorphism φ(a)=a, φ(b)=ab and φ(c)=bcb. The mapping torus uses the convention tut⁻¹=φ(u); the final b in bcb is positive.

There is a finite locally CAT(0) unit cube complex Z of dimension three, with cell counts (1,7,8,2) and fundamental group isomorphic to the full group F⋊φZ. Its universal cover carries a free, proper and cocompact action of that exact group by cubical isometries.

The finite quotient uses the general cube-complex convention allowing isometric boundary-face identifications. The stated cell counts are before subdivision; the CAT(0) universal cover satisfies the stricter embedded-cube convention.

Why the result holds

A base built from two squares admits a cubical involution. A second two-square complex has infinite cyclic fundamental group and two compatible local-isometry maps into the base.

Attaching the ends of its product with an interval gives the finite quotient. The links are checked with signed edge directions and all repeated corner occurrences retained; the unique vertex link is an actual flag simplicial complex.

Reversible presentation changes identify the entire fundamental group, including the positive final b in the automorphism. Standard cubical geometry then gives the CAT(0) universal cover and the proper cocompact deck action.

A separate argument explains qualitative existence using earlier WPD, Morse and cubical-completion theorems. Those prior tools do not supply the displayed small cell counts.

Scope and limitations

The result concerns this exact marked automorphism. The workshop summary also discusses a related example with b↦aba; that formula is distinct from the b↦ab target here.

No minimum dimension, minimum cell count, virtual specialness or theorem for a whole family of free-by-cyclic groups is claimed.

The original quotient permits boundary identifications. A stricter cubical quotient can be obtained by subdivision, but the counts (1,7,8,2) then change.

The explicit construction is presented with credit to earlier geometric tools. The paper makes no historical-firstness or absolute-novelty claim, and an archived problem statement does not establish current worldwide openness.

Status and AI use

AI tools were used extensively in discovery, drafting and internal adversarial checking. Internal AI reviews and finite computations are not external human peer review or proof-assistant certification.

The manuscript and Zenodo publication-date metadata are dated 12 October 2026. Zenodo records this version as created on 13 October 2026 (UTC). The displayed date is the manuscript date, not verified public availability on that date. Neither date establishes first priority. Version 1.0 is the Zenodo version; the manuscript does not print a version number.

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Collaborate on this result

I welcome substantive collaboration from researchers in geometric group theory and CAT(0) cube complexes on independent verification, comparison with existing cubulations and development toward journal review. Coauthorship would reflect actual research contributions and mutual agreement.

Authorship and responsibilities would be discussed in relation to the substantive contributions made.

Cite this paper

Alec Kriebel. (2026). A two-cube cubulation of the AIM free-by-cyclic example (Zenodo version 1.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23328912

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Background and prior work

• AIM Problem Lists — Rigidity properties of free-by-cyclic groups, Question 5.2: The archived question, attributed to Rylee Lyman, gives the exact a↦a, b↦ab, c↦bcb formulation and asks for a proper cocompact CAT(0) cubical action.
• AIM — Rigidity properties of free-by-cyclic groups: workshop summary: Page 2 discusses a related b↦aba example. It should be kept distinct from the marked group in Question 5.2.
• Leary — A metric Kan–Thurston theorem: Appendix C supplies the flag-link criterion for general cube complexes and the embedded-cube and subdivision conclusions.
• Minasyan and Osin — Acylindrical hyperbolicity of groups acting on trees: Corrected Corollary 4.3 supplies the WPD step in the separate qualitative existence argument.
• Dahmani, Guirardel and Osin — Hyperbolically embedded subgroups and rotating families: Theorem 6.8 and Corollary 2.9 turn the relevant loxodromic WPD element into a hyperbolically embedded virtually cyclic subgroup.
• Sisto — Quasi-convexity of hyperbolically embedded subgroups: Theorem 1 supplies the Morse property used to obtain word-metric quasiconvexity.
• Ben-Zvi, Kropholler and Lyman — Folding-like techniques for CAT(0) cube complexes: Finite-completion results provide another finite cyclic local isometry in the qualitative existence route.
• Hagen and Wise — Cubulating mapping tori of some polynomial growth free group automorphisms: Remark 3.7 discusses local-isometry gluing; the general free-action result is distinct from the proper cocompact conclusion established here.